a new method for estimating maximum power … · operating point to the “ nose point ” on a...
TRANSCRIPT
Bernie Lesieutre Dan Molzahn
University of Wisconsin-Madison
A New Method for Estimating Maximum Power Transfer and Voltage
Stability Margins to Mitigate the Risk of Voltage Collapse
PSERC Webinar, October 15, 2013
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Voltage Stability
Static: - Loading, Power Flows - Local Reactive Power Support
Dynamic:
- Component Controls - Network Controls
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OVERVIEW: Static Voltage Stability Margin
Power
Voltage, V Margin
A common voltage stability margin measures the distance from a post-contingency operating point to the “nose point” on a power-voltage curve.
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OVERVIEW: Static Voltage Stability Margin
Power
Voltage, V Margin (negative)
It’s even possible that a normal power flow solution won’t exist, post contingency.
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ISSUES
Power
Voltage, V Margin
1. We don’t know the values on the curve. 2. We don’t know whether a post-contingency operating point
exists!
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Typical Approach
• Run post-contingency power flow. This may or may not converge.
• If successful, determine nose point: use a sequence of power flows with increasing load, or a continuation power flow.
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Our Approach
Cast as an optimization problem: • Minimize the controlled voltages while a solution exists. (Claim: a solution exists.)
• Exploit the quadratic nature of the power flow equations to directly obtain
Traditional Voltage Stability Margin even when the margin is negative.
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Advantages
• Eliminates need for repeated solutions (multiple power flows, continuation power flows) • Often offers provably globally optimal results • Works when the margin is negative, i.e. when there isn’t a solution.
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OVERVIEW: Static Voltage Stability Margin
Power
Voltage, V Margin
Optimal Solution
Vopt
V0
P0 Pmax
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Power Flow Equations: Review
Electrical Network
I = YV Y = G+jB
(ID1+jIQ1)
(IDN+jIQN)
(VD1+jVQ1)
(VDN+jVQN)
Current “Injection” equations in Rectangular Coordinates:
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Power Flow Equations: Review
Electrical Network
I = YV Y = G+jB
(P1+jQ1)
(PN+jQN)
(VD1+jVQ1)
(VDN+jVQN)
Power “Injection” equations in Rectangular Coordinates:
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Power Flow Equations
The power flow equations are quadratic in voltage variables:
For reference, power engineers almost always express these equations in voltage polar coordinates:
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Classical Power Flow Constraints
Bus Type Specified Calculated
PQ (load) P, Q V, δ
PV (generator) P, V Q, δ
Slack V, δ = 0º P, Q
The “controlled voltages” are the problem-specified slack and generator voltage magnitudes. Locking the controlled voltages in constant proportion, and allowing them to scale by α, we claim a power solution exists for any power flow injection profile. (subject to the unimportant small print.)
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Optimization Problem • Modify the power flow formulation to • Slack bus voltage magnitude unconstrained
• PV bus voltage magnitudes scale with slack bus voltage
• Minimize slack bus voltage
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This optimization problem can be solved many different ways…
We’ve been using the convex relaxation formulation for the power flow equations (Lavaei, Low) because we really want (provably) the minimum solution.
• The problem has a feasible solution • The optimization using the convex relaxation can be
solved for a global minimum (and hopefully a feasible power flow solution).
Optimization Problem
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Relaxed Problem Formulation
In Rectangular coordinates, define
Then, power flow equations can be written in the form
where
Which is a rank one matrix by construction.
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Relaxed Problem Formulation
The convex relaxation is introduced by relaxing the rank of W. With that, we pose the following convex optimization
problem:
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Bonus result concerning solution to the power flow equations
• The existence of a power flow solution requires
• Necessary, but not sufficient, condition for existence
• Conversely, no solution exists if
• Sufficient, but not necessary, condition for non-existence
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Controlled Voltage Margin
• A controlled voltage margin to the solvability boundary
• Upper bound (non-conservative)
• No power flow solution exists for
• Increasing the slack bus voltage (with proportional increases in PV bus voltages) by at least is required for solution.
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Power Injection Margin • Uniformly scaling all power injections scales
• Uniformly scale power injections until
• Corresponding gives a power flow voltage stability margin in the direction of uniformly increasing power injections at constant power factor.
• indicates that no solution exists for the original power flow problem
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Power Injection Margin
The power injection margin answers the question
For a given voltage profile, by what factor can we change our power injections (uniformly at all buses) while still potentially having a solution?
Answer:
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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1.1IEEE 14-Bus Continuation Trace: Bus 5
Injection Multiplier
V5
Controlled Voltage Margin
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1IEEE 14-Bus Continuation Trace: Bus 5
Injection Multiplier
V5
Power Injection Margin
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IEEE 14 Bus System Injection
Multiplier
Newton-Raphson
Converged?
dim(null(A)
1.000 Yes 1.06 0.5261 2 2.000 Yes 1.06 0.7440 2 3.000 Yes 1.06 0.9112 2 4.000 Yes 1.06 1.0522 2 4.010 Yes 1.06 1.0535 2 4.020 Yes 1.06 1.0548 2 4.030 Yes 1.06 1.0561 2 4.040 Yes 1.06 1.0575 2 4.050 Yes 1.06 1.0588 2 4.055 Yes 1.06 1.0594 2 4.056 Yes 1.06 1.0595 2 4.057 Yes 1.06 1.0597 2 4.058 Yes 1.06 1.0598 2 4.059 Yes 1.06 1.0599 2 4.060 No 1.06 1.0601 2 4.061 No 1.06 1.0602 2 4.062 No 1.06 1.0603 2 4.063 No 1.06 1.0605 2 4.064 No 1.06 1.0606 2 4.065 No 1.06 1.0607 2 5.000 No 1.06 1.1764 2
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
IEEE 14-Bus Continuation Trace: Bus 5
Injection Multiplier
V5
Voltage and Power Injection Margins
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0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
IEEE 118-Bus Continuation Trace: Bus 44
Injection Multiplier
V44
Controlled Voltage Margin
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0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
IEEE 118-Bus Continuation Trace: Bus 44
Injection Multiplier
V44
Power Injection Margin
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IEEE 118 Bus System Injection
Multiplier
Newton-Raphson
Converged?
(lower bound)
dim(null(A)
1.00 Yes 1.035 0.5724 4 1.50 Yes 1.035 0.7010 4 2.00 Yes 1.035 0.8095 4 2.50 Yes 1.035 0.9050 4 3.00 Yes 1.035 0.9914 4 3.15 Yes 1.035 1.0159 4 3.16 Yes 1.035 1.0175 4 3.17 Yes 1.035 1.0191 4 3.18 Yes 1.035 1.0207 4 3.19 No 1.035 1.0223 4 3.20 No 1.035 1.0239 4 3.21 No 1.035 1.0255 4 3.22 No 1.035 1.0271 4 3.23 No 1.035 1.0287 4 3.24 No 1.035 1.0303 4 3.25 No 1.035 1.0319 4 3.26 No 1.035 1.0335 4 3.27 No 1.035 1.0351 4 3.28 No 1.035 1.0366 4 3.29 No 1.035 1.0382 4 4.00 No 1.035 1.1448 4
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0 0.5 1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
1.2
IEEE 118-Bus Continuation Trace: Bus 44
Injection Multiplier
V44
Voltage and Power Injection Margins
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Alternate Power Injection Profiles Power injection margin is in the direction of a uniform,
constant-power-factor injection profile
We can alternatively specify any profile that is a linear function of powers and squared voltages
‒ However, insolvability condition is not necessarily valid
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Reactive Power Limits Previous work models generators
as ideal voltage sources
Detailed models limit reactive outputs
‒ Limit-induced bifurcations
Two approaches to modeling these limits:
‒ Mixed-integer semidefinite programming
‒ Infeasibility certificates using sum of squares programming
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Voltage Magnitude
Rea
ctiv
e P
ower
Reactive Power versus Voltage Magnitude Characteristic
V∗
Qmax
Qmin
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MISDP Formulation
0
Voltage Magnitude
Rea
ctiv
e P
ower
Reactive Power versus Voltage Magnitude Characteristic
V∗
Qmin
Qmax
Model reactive power limits using binary variables
Insolvability
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MISDP Formulation
0
Voltage Magnitude
Rea
ctiv
e P
ower
Reactive Power versus Voltage Magnitude Characteristic
V∗
Qmin
Qmax
Model reactive power limits using binary variables
Insolvability
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MISDP Formulation
0
Voltage Magnitude
Rea
ctiv
e P
ower
Reactive Power versus Voltage Magnitude Characteristic
V∗
Qmin
Qmax
Model reactive power limits using binary variables
Insolvability
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MISDP Formulation
0
Voltage Magnitude
Rea
ctiv
e P
ower
Reactive Power versus Voltage Magnitude Characteristic
V∗
Qmin
Qmax
Model reactive power limits using binary variables
Insolvability
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Reactive Power Limits Results
1 1.1 1.2 1.3 1.40.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.02514-Bus System Power vs. Voltage
PQ Bus Injection Multiplier
V5
P-V Curve
ηmax
Infeasibility Certificate Found
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Conclusions • Cast the problem of computing voltage stability margins as
an optimization problem – to minimize the slack bus voltage.
• Calculated voltage stability margins – power injection/flows, and controlled voltages.
• Tested with numerical examples
Advantages:
• Eliminates repeated solution (multiple power flows, continuation power flows)
• Often offers provably globally optimal results
• Works when the margin is negative, i.e. when there isn’t a solution.
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Related Publications
[1] B.C. Lesieutre, D.K. Molzahn, A.R. Borden, and C.L. DeMarco, “Examining the Limits of the Application of Semidefinite Programming to Power Flow Problems,” 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2011, pp.1492-1499, 28-30 Sept. 2011.
[2] D.K. Molzahn, J.T. Holzer, and B.C. Lesieutre, and C.L. DeMarco, “Implementation of a Large-Scale Optimal Power Flow Solver Based on Semidefinite Programming,” To appear in IEEE Transactions on Power Systems.
[3] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, “An Approximate Method for Modeling ZIP Loads in a Semidefinite Relaxation of the OPF Problem,” submitted to IEEE Transactions on Power Systems, Letters.
[4] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Global Optimality of Solutions to the Optimal Power Flow Problem,” to appear in IEEE Transactions on Power Systems, Letters.
[5] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, “A Sufficient Condition for Power Flow Insolvability with Applications to Voltage Stability Margins,” IEEE Transactions on Power Systems, Vol 28, No. 3, pp. 2592-2601.
[6] D.K. Molzahn, V. Dawar, B.C. Lesieutre, and C.L. DeMarco, “Sufficient Conditions for Power Flow Insolvability Considering Reactive Power Limited Generators with Applications to Voltage Stability Margins,” presented at Bulk Power System Dynamics and Control - IX. Optimization, Security and Control of the Emerging Power Grid, 2013 IREP Symposium, 25-30 Aug. 2013.
[7] D.K. Molzahn, B.C. Lesieutre, and C.L. DeMarco, “Investigation of Non-Zero Duality Gap Solutions to a Semidefinite Relaxation of the Power Flow Equations,” To be presented at the Hawaii International Conference on System Sciences, January 2014.
Conclusion
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For lossless systems: 1. Show a solution exists for zero power injections at
PQ buses and zero active power injection at PV buses, for α = 1.
2. Use implicit function theorem to argue that perturbations to zero power injections solutions also exist. Specifically choose one in the direction of desired power injection profile.
3. Exploit the quadratic nature of power flow equations to scale voltages and power to match injection profile.
Feasibility
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Easy: Construct a solution. • Open Circuit PQ buses for zero power, zero
current injection. • Use Ward-type reduction to eliminate PQ buses.
(not really necessary, but clean) small print
• Choose uniform angle solution for all buses. • Directly use reactive power flow equation to
calculated the reactive power injections at generator buses.
Feasibility: Zero Power Injection Solution
0
0
0
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Note: Zero Power Injection Solutions for Lossy Systems
• Not all systems have a zero-power injection solution
• Ability to choose θ such that PPV = 0 depends on
• Ratio of VPV to Vslack
• Ratio of g to b
• Systems with small resistances and small voltage magnitude differences are expected to have a zero power injection solution
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A nearby non-zero solution exists provided the Jacobian is nonsingular. For a connected
lossless system at the zero-power injection solution, the appropriate Jacobian is nonsingular, generically.
small print
Nearby Solutions
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Exploit the quadratic nature of power flow equations to scale voltages to match desired power profile:
Feasible Solution
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Infeasibility Certificates
Guarantee that a system of polynomial is infeasible
Positivstellensatz Theorem
If such that
then the system of polynomials has no solution
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Power Flow in Polynomial Form Power Injection and Voltage Magnitude Polynomials
Reactive Power Limit Polynomials
0
Voltage Magnitude
Rea
ctiv
e P
ower
Reactive Power versus Voltage Magnitude Characteristic
V∗
Qmax
Qmin